In my paper, I had argued from the fact of "persistent and apparently intractable disaggrement on foundational questions" in moral theory to the conclusion that we ought to be skeptical about the existence of moral facts or properties. JCD claims that we find the same kind of disagreement in mathematics. This might, of course, lead us to skepticism about mathematical facts or it might lead us to worry about this strategy of argument. JCD doesn't take a position on that issue here. He wants, in his first set of comments, to make the case that the situation, when it comes to disagreement, is the same in math as I claim it is in ethics. (His second set of comments raise a different set of issues, to which I'll turn in a second post.)

As an example of "persistent" disagreement in mathematics, JCD notes that, "There has been disagreement over the axioms for set-theory since their formulation."

As evidence that this disagreement is "apparently intractable," JCD notes that "there is not merely disaggrement over the truth-values of mathematical sentences, but also disagreement over what would count as evidence for those sentences' truth or falsity." JCD gives the examples of disagreement over the Choice and Replacement axioms.

As evidence that this disagreement has a "foundational character," JCD notes that disagreement over the axioms for set-theory is "epistemically foundational" for mathematics, since these are "the only serious candidates which might be thought to justify axioms which imply them." Such disagreement is also "metaphysically foundational" since "the axioms of our 'explanatorily' fundamental theory, set theory" are the metaphysical foundation of mathematics.

JCD has the significant advantage here of knowing a lot more mathematics than I do, though I hope to hear from readers also conversant in the underlying mathematical debates. I don't want to take issue with the question of whether the putative disagreements in questions are "foundational" (other readers are welcome to do so); I do want to pose some questions about whether they are "persistent" and "apparently intractable."

Set theory dates from the late 19th-century, so has been subject to about 140 years of development and dispute by mathematicians. How does the state of disagreement today compare to 100 years ago? To 50 years ago? To 25 years ago? Reading the SEP entry on set theory , one is left with the impression of a progressive discipline with gradual agreement on many basic ideas. Why is this? Could any entry on the foundations of morality read like the SEP entry on set theory?

The latter, purely sociological, observations bear on the question of intractability. It appears that many foundational issues in set theory have been resolved since the 1870s. (Is there any foundational issue about the criteria of right action that has been resolved since the 1870s? Or since the 1670s?) Evidence of 'intractability' has partly to do with persistence, but partly to do with the terms in which disagreement is carried out. Foundational debates in ethics devolve into clashing intuitions and accusations of moral corruption and obtuseness rather quickly! What are the terms on which apparently 'intractable' debates about set theory are carried out?

JCD makes the interesting sociological point that "the correlation between relevant mathematicians' views and thsoe of their mentor is impossible to miss." How long does that pattern last? Over multiple generations? Or do we find that the student of X often comes to reject the views of his teacher's teacher?

I don't presuppose answers to these questions. But their answers might well suggest which horn of JCD's dilemma we should embrace (i.e., mathematical skepticism or skepticiam about the argument from disagreement).

I should note that I don't quite understand JCD's reference, near the end of his first set of remarks, to the fact that fictionalists in philosophy of mathematics deny simple arithmetical truths. The argument from moral disagreement appeals to first-order disagreement in ethical theory about the criteria of right action, the nature of moral goodnes, and the relative priority of rightnes and goodness in the evaluatoin of actions and persons, among other considerations. It does not depend on claims about the metaphysics, epistemology, or semantics of these judgments. JCD is obviously skeptical about drawing the line between the meta- and first-order disagreements in mathematics. I would like some further explanation of why, and whether there is any reason for that skepticism to carry over to the meta-ethical case.

4 comments:

Your sociological questions about the “state of disagreement” in set theory over the years brought to mind Kuhn’s analysis of this state in science. When Nietzsche’s work is compared with Kuhn’s analysis, I found that there is an interesting agreement on the tempo of revaluations of value that could contribute to this discussion as well as your review of Shaw’s book.

Nietzsche criticizes a “TROPICAL TEMPO” (BGE 242) were there are “savagely opposing” interests that undermine agreement; he also criticizes a “prestissimo tempo” (WP 71) where people change their mind quickly. These tempos undermine “tension of the bow” that enables societies to aim “at the furthest goals” (BGE Introduction). Implicit in these criticisms and the bow and arrow analogy is the suggestion that a society needs to manifest widespread agreement on goals and be able to persist until they are fulfilled in order to grow in power.

The largo and symphonic tempo he implicitly associates with the growth of power is directly parallel to Kuhn’s discussion of paradigms in The Structure of Scientific Revolutions. They are described as frameworks of understanding that "attract an enduring group of adherents" (Kuhn 1962, 10) and guide the thinking and activity of this group over an extended period of time—centuries in many cases. Consequently, Kuhn writes that normal science is "the activity in which most scientists inevitably spend almost all their time..." (Kuhn 1962, 5).

Nietzsche’s description of the value of this largo, symphonic tempo speaks to the question of a "stable political authority" that Shaw addresses without ignoring his moral anti-realism. Values are indeed created; but to affirm the growth of life we need to create them in a tempo that is neither tropical nor prestissimo: “the thing that is to be avoided above everything is further experimentation—the continuation of the state in which values are fluent, and are tested, chosen and criticized ad infinitum” (A 57). It appears to me that Nietzsche, like Kuhn, recognized the value of paradigms and the organized activity to which they give rise. The largo, symphonic tempo they describe enables societies to “reap as rich and as complete a harvest [from their values] as possible” (A 57).

Hi, Brian. Thanks so much for your thoughtful response to my first set of comments.

Before responding to it in turn, let me make a clarificatory point.

Midway through your response you write,

“…JCD notes that disagreement over the axioms for set-theory is "epistemically foundational" for mathematics, since these are "the only serious candidates which might be thought to justify axioms which imply them."

What I said was:

“There is disagreement over epistemically foundational propositions in mathematics since there is disagreement over the only serious candidates for these – namely, axioms, and theorems which might be thought to justify axioms which imply them.”

This is different from the position I take you to be attributing to me. I want to distance myself from the view that epistemically basic mathematical claims are limited to axioms. Some axioms, such as Replacement, seem to be endorsed on a basis that is akin to the basis on which fundamental hypotheses in empirical science are endorsed. We endorse Replacement because it allows us to prove things that we think are true – namely, that ordinals as great as or greater than omega + omega exist – not because it is intuitively obvious. Indeed, it is hard to see how anyone could find Replacement to be intuitively obvious given that it easily implies such dramatic results as that there is an ordinal greater than all f(x), where f(0) = Aleph_0 and f(x+1) = Aleph_f(x) for all natural numbers, x.

Nevertheless, I have no vested interest in this issue.

You write,

“Reading the SEP entry on set theory , one is left with the impression of a progressive discipline with gradual agreement on many basic ideas….It appears that many foundational issues in set theory have been resolved since the 1870s.”

I agree that the relevant SEP entry leaves on with this impression. However, I think that this can be explained in a way that is consistent with my thesis.

First, when I say that “there is radical disagreement in mathematics”, I am not saying that most mathematicians prefer to work in a wide array of theories. A mathematician might think that Quine’s NF is more probably true than ZFC, but prefer to work in ZFC for most purposes because it is far more intuitive. However, I suspect that Jech is not distinguishing the claim that there is a great deal of agreement as to what mathematical theories are preferable to work in and the claim that there is a great deal of agreement as to what mathematical theories are probably true.

Second, Jech is clearly not writing the relevant article from the realist perspective that is relevant here. For instance, with regard to the Axiom of Choice, he writes, “The legitimate question is whether the Axiom of Choice is consistent, that is whether it cannot be refuted from other axioms…” Obviously this is not “the” legitimate question if one thinks that there is a unique mathematical truth. There is the additional question of whether the Axiom of Choice is true. Establishing its consistency with ZF doesn’t answer that (both Choice and its negation are consistent with ZF).

However, if one does think that the only legitimate question with respect to a mathematical hypothesis is that of whether it is consistent with relevant axioms (as a formalist might), then of course she will think that there has been dramatic progress in set theory. We have learned a great deal about what follows from, is inconsistent with, and is independent of, what in mathematics. But one could have perfect knowledge of this without having any knowledge of what mathematical claims are true. By way of illustration, suppose that we formalized the myriad of ethical theories, and gained complete knowledge of what follows from, is inconsistent with, and independent of, what in ethics. There would obviously remain the question of what ethical claims are true.

“Foundational debates in ethics devolve into clashing intuitions and accusations of moral corruption and obtuseness rather quickly! What are the terms on which apparently 'intractable' debates about set theory are carried out?”

I think this varies about as much as it varies in ethics, taking into account the peculiar background of mathematicians. In publications, the relevant debates tend to meet a high standard of professionalism, but, on listserves and blogs, such debates fairly often turn ugly. For example, there have been various unfortunate “discussions” of relevant matters on the FOM – enough so that the issue of etiquette has been explicitly broached with the subscribers as a whole on more than one occasion.

“JCD makes the interesting sociological point that "the correlation between relevant mathematicians' views and those of their mentor is impossible to miss." How long does that pattern last?”

Hi again, Brian. I noticed that I failed to respond to your last point, regarding the relevance of fictionalism to arguments from disagreement. You write,

“The argument from moral disagreement appeals to first-order disagreement in ethical theory about the criteria of right action….JCD is obviously skeptical about drawing the line between the meta- and first-order disagreements in mathematics. I would like some further explanation of why, and whether there is any reason for that skepticism to carry over to the meta-ethical case.”

I brought up fictionalists for two reasons.

The first is that many seem to work with a picture of mathematics according to anyone who rejects the likes of “2 + 2 = 4” must be crazy – that there are no reasons that might compel a sensible person to reject such propositions. Since it is not the case that anyone who rejects the most banal of purported moral truths must be crazy, one might be left with the impression that there is an important disanalogy between mathematics and ethics in this regard. I just wanted to emphasize that there is not.

I also brought up fictionalists, as you suggest, because I doubt that there is an interesting first-order/second-order distinction in math (I don’t think that I have any dialectical need to show that this skepticism carries over to ethics, though I will suggest that it does).

One reason for this is that I have trouble seeing why mathematical disagreement motivated only by first-order considerations would be more problematic for mathematical realism than disagreement motivated by second-order considerations. If mathematical disagreement has bearing on mathematical realism, then it seems to me to be when such disagreement is among “cognitively virtuous” folks -- independent of whether those folks base their mathematical beliefs on first-order or second-order considerations.

The reason for doubt that I’d like to develop here, however, is that intuitively first-order mathematical claims are systematically based on intuitively second-order ones.

The peculiar epistemic question with respect to any domain, F, is not how we know our F-theory, given that we know the F-propositions on which we ultimately (epistemically) base that theory. It is how we know those F-propositions themselves. For example, if axioms are (epistemically) basic in mathematics, the peculiar epistemic question with respect to mathematics is not how we know mathematical theorems, given that we know mathematical axioms. It is how we know mathematical axioms themselves.

Part of the answer to the peculiar epistemic question with respect to mathematics is that we know basic mathematical propositions on the basis of considerations about the intuitive character of the mathematical universe, the extent to which mathematical objects depend on cognition, and the role of mathematics in empirical science. Classic arguments for and against intuitively basic mathematical propositions practically always appeal to such considerations. Consider Descartes’ arguments against imaginary numbers, Newton’s arguments for the existence of infinitesimals, Cantor’s arguments for the legitimacy of the transfinite, Poincare’s arguments against impredicative definitions, Aczel’s arguments against the well-foundedness of the set-theoretic universe, and current arguments for large cardinals and various “solutions” to the continuum problem.

Considerations about the intuitive character of the mathematical universe, the extent to which mathematical objects depend on cognition, and the role of mathematics in empirical science, are each second-order. Take considerations about the extent to which mathematical objects depend on cognition. These figured centrally into Poincare’s arguments against impredicative definitions. Given that mathematical definitions serve a role akin to recipes for mathematical construction, how could a definition of a mathematical object in terms of a set to which it belongs be coherent? This question is deeply relevant what first-order mathematical claims we should endorse (induction? mean-value theorem? Cantor’s theorem?). But it is also second-order if anything is.

I think that parallel things could be said with respect to the first-order/second-order distinction in ethics, aesthetics, (empirical) science, religion, and law. Intuitively second-order considerations pervade all subjects, in that one cannot hold significant array of justified beliefs on those subjects without partly basing them on intuitively second-order beliefs. A potential upshot of this is that arguments from disagreement against corresponding brands of realism may be reduced to arguments from disagreement against philosophical realism.

About Me

Brian Leiter is Karl N. Llewellyn Professor of Jurisprudence and Director of the Center for Law, Philosophy, & Human Values at the University of Chicago. He works on a variety of topics in moral, political, and legal philosophy. His current Nietzsche-related work concerns Nietzsche's theory of agency and its intersection with recent work in empirical psychology; Nietzsche's arguments for moral skepticism; and the role of naturalism in Nietzsche's philosophy.